Polarization states encoded by phase modulation for high bit rate quantum key distribution

Physics Letters A 358 (2006) 386–389 www.elsevier.com/locate/pla

Polarization states encoded by phase modulation for high bit rate quantum key distribution Xiaobao Liu ∗ , Zhilie Tang, Changjun Liao, Yiqun Lu, Feng Zhao, Songhao Liu School for Physics and Telecom Engineering, South China Normal University, Guangzhou, China School for Information and Optoelectronic Science and Engineering, South China Normal University, Guangzhou, China Received 16 January 2006; received in revised form 28 April 2006; accepted 23 May 2006 Available online 5 June 2006 Communicated by P.R. Holland

Abstract We present implementation of quantum cryptography with polarization code by wave-guide type phase modulator. At four different low input voltages of the phase modulator, coder encodes pulses into four different polarization states, 45◦ , 135◦ linearly polarized or right, left circle polarized, while the decoder serves as the complementary polarizers. © 2006 Elsevier B.V. All rights reserved. PACS: 03.67.Dd Keywords: Polarization states; Phase modulation; Polarizing M–Z interferometer; Quantum key distribution

Quantum key distribution (QKD) over free-space path [1–7] has attracted much attentions in recent years, since absolute security of key generation and exchange based on the quantum mechanics and its applications in conventional fiber communication system have been practically demonstrated. To date, free-space QKD has been demonstrated over outdoor path up to 10 km in daylight [5] and 23 km at night [6]. One of the main concerns is the key transfer speed. The increase in speed could potentially make free-space quantum cryptography practical for applications such as video and high speed communication. The speed of key transfer can be increased upon improvement in coding and detection technique. It has been widely recognized that polarization-code is insensitive to air turbulence in comparison with phase-code which has been extensively used in fiber quantum key distribution system where birefringence of the standard telecom fiber causes the polarization-code transmission unstable. The existing weak pulse implementations of BB84 [8] or B92 [9] protocols with polarization codes can be divided into two categories: one uses Pockels cells; the other

* Corresponding author.

E-mail address: [email protected] (X. Liu). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.05.068

contains of a set of lasers and polarizers. The former implementations need only one laser. Its coding speed, however, is slow due to the high driving-voltage for the Pockels cells. At Bob’s side, all implementations use sets of analyzers to decode the polarization states. A protocol called phase-modulation polarization coding [10] was suggested to encode and decode different polarization states by phase modulation. Both its transmitter and the receiver are based on the structure of polarizing M–Z interferometers [11]. Here we describe the implementation of quantum cryptography with polarization code by wave-guide type LiNbO3 phase modulator which allows high-speed operation at several voltages. The schematic of Alice is depicted as Fig. 1. Define vertically and horizontally polarized states that go through PBS1 into arm I as |V , 1 and |H, 1 respectively, and those that go through PBS2 into arm II as |V , 2 and |H, 2 respectively. Paths outside polarizing interferometer (that is, all paths except arm I and arm II) are marked path 0. |V , 0 and |H, 0 denote respectively the vertically and horizontally polarized states that go through path 0. The single photon source is provided by attenuated laser pulse. The initial polarization of the photon is controlled by polarization controller to produce a linear polarization at the direction 45◦ inclined to the horizon-

X. Liu et al. / Physics Letters A 358 (2006) 386–389

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a λ/2 retarder Wˆ θ = cos 2θ|H, 0H, 0| + sin 2θ |H, 0V , 0| + sin 2θ |V , 0H, 0| − cos 2θ |V , 0V , 0|. Fig. 1. Schematic of the Alice. Two polarizing beam splitters (PBS) and polarization-maintaining fibers make up a polarizing M–Z interferometer, with a wave-guide type phase modulator (PM) at arm II. PC, polarization controller. T, transmitting optics. La, laser.

(4)

θ stands for the direction of its fast-axis inclined to the horizontal. The qubit finally arrives at PBS3 Pˆ = |H, 1H, 0| + |V , 2V , 0|.

(5)

Bob’s entire set is Bˆ = P Wθ M = cos 2θ |H, 1H, 0| + sin 2θ |V , 2H, 0| + sin 2θ eiΦ |H, 1V , 0| − cos 2θ eiΦ |V , 2V , 0|.

(6)

45◦

Fig. 2. Schematic of the Bob. λ/2 is a λ/2 retarder (pigtailed in each ends). R, receiving optics. Table 1 Different output polarized states at Alice side Φ 0 π 3π/2 π/2

|out √ √2/2(|H, 0 + |V , 0) √2/2(|H, 0 − |V , 0) √2/2(|H, 0 − i|V , 0) 2/2(|H, 0 + i|V , 0)

Output polarization state 45◦ linearly polarized |45 135◦ linearly polarized |135 left-circular polarized |L right-circular polarized |R

tal so that the input to the polarization beam splitter 1 (PBS1) can be expressed as √ √ |in45 = 2/2|H, 0 + 2/2|V , 0. (1) The qubit then goes through the polarizing M–Z interferometer which is described by Mˆ = |H, 0H, 0| + eiΦ |V , 0V , 0|,

(2)

where Φ is the phase difference introduced by the phase modulator. The output polarization states of the M–Z interferometer can be given by √ √ iΦ ˆ 2/2|V , 0. |out = M|in (3) 45 = 2/2|H, 0 + e Eq. (3) suggests that output polarization state is only determined by Φ. When Φ is set at 0, π, 3π/2 and π/2, output states of the interferometer are 45◦ linearly polarized |45, 135◦ linearly polarized |135, left circle polarized |L and right circle polarized |R, respectively (see Table 1). These four states are our main concern. They are the four different states required in BB84 protocol for they belong to two nonorthogonal bases, H /V basis (|45 and |135) and circular basis (|L and |R). In this case, Alice easily codes a qubit to one state either of H /V basis and circular basis, only by selecting its corresponding Φ. Bob is mainly comprised of a polarizing M–Z interferometer, a λ/2 retarder and a PBS showed in Fig. 2. The qubit travels through PC which is used to recover the initial polarization states by compensating for birefringence effect induced by optical fiber and goes through Bob’s polarizing M–Z interferometer (described by Eq. (2)). Then it pass through

To ensure same possibility that Bob plays role as linearly polarizer, 135◦ linearly polarizer, left-circular polarizer or rightcircular polarizer, θ must be set to π/8 √ 2 ˆ B= |H, 1H, 0| + |V , 2H, 0| 2  + eiΦ |H, 1V , 0| − eiΦ |V , 2V , 0| . (7) When PM is selected among 0, π , −π/2 and π/2, Bob is described as follows: √ 2 ˆ B0 = |H, 1H, 0| + |V , 2H, 0| 2  + |H, 1V , 0| − |V , 2V , 0| (Φ = 0), (8) √  2 Bˆ π = |H, 1H, 0| + |V , 2H, 0| 2  (9) − |H, 1V , 0| + |V , 2V , 0| (Φ = π), √ 2 |H, 1H, 0| + |V , 2H, 0| Bˆ 3π/2 = 2  − i|H, 1V , 0| + i|V , 2V , 0| (Φ = 3π/2), (10) √ 2 |H, 1H, 0| + |V , 2H, 0| Bˆ π/2 = 2  + i|H, 1V , 0| − i|V , 2V , 0| (Φ = π/2). (11) Eqs. (8)–(11) suggest four most important roles that Bob plays: 45◦ linearly polarizer Bˆ 0 , 135◦ linearly polarizer Bˆ π , left-circular polarizer Bˆ 3π/2 and right-circular polarizer Bˆ π/2 when Φ is 0, π , 3π/2 and π/2, respectively. Take 45◦ linearly polarizer Bˆ 0 for example. If Alice sends a |45 qubit, that is, |45 input at Bob’s side, Bob’s output is Bˆ 0 · |45 = |H, 1. This suggests that a |H  qubit would come out of output 1 of the PBS3 with certainty. If Alice sends a |135 qubit, Bob’s output is Bˆ 0 · |135 = |V , 2 which suggests that |V  qubit would come out of output 2 of the PBS3 with certainty. However, if Alice sends a |L qubit, Bob’s output is Bˆ 0 · |L = 1/2[(1 − i)|H, 1 + (1 + i)|V , 2]. This suggests qubit would come out of either of PBS3’s outputs with 50% possibility respectively. It turns out that Bob is equivalent to a 45◦ linearly polarizer when Φ is 0, distinguishing between |45 and |135 input with 100% possibility and between |L and |R with 50% possibility. Other roles that Bob plays are expatiated on in Table 2. In conclusion, Bob makes two nonorthogonal bases

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X. Liu et al. / Physics Letters A 358 (2006) 386–389

Table 2 Bob function as 45◦ linearly polarizer, 135◦ linearly polarizer, left-circular polarizer or right-circular polarizer when Φ is set at 0, π , −π/2 or π/2. |H, 1 means the qubits come out from port 1 for certain with horizontal polarization and |V , 2 means the qubits come out from port 2 for certain with vertical polarization. The output state 1/2[(1 − i)|H, 1 + (1 + i)|V , 2] and 1/2[(1 + i)|H, 1 + (1 − i)|V , 2] means the possibility qubits come out from port 1 and port 2 is both 50%, that is, a random outcome Bob’s PM (Φ) Bob’s output

0

π

3π/2

π/2

45◦ linearly polarized |45

|H, 1

|V , 2

1/2[(1 − i)|H, 1 + (1 + i)|V , 2]

1/2[(1 + i)|H, 1 + (1 − i)|V , 2]

135◦ linearly polarized

|V , 2

|H, 1

1/2[(1 + i)|H, 1 + (1 − i)|V , 2]

1/2[(1 − i)|H, 1 + (1 + i)|V , 2]

left-circular polarized

1/2[(1 − i)|H, 1 + (1 + i)|V , 2]

1/2[(1 + i)|H, 1 + (1 − i)|V , 2]

|V , 2

|H, 1

right-circular polarized

1/2[(1 + i)|H, 1 + (1 − i)|V , 2]

1/2[(1 − i)|H, 1 + (1 + i)|V , 2]

|H, 1

|V , 2

Bob’s equivalent

45◦ linearly polarized polarizer

135◦ linearly polarized polarizer

left-circular polarized polarizer

right-circular polarized polarizer

basis

diagonal basis

diagonal basis

circular basis

circular basis

input state

measurements, H /V basis and circular basis, which meets the BB84 protocol’s requirement. Our experiment first concentrates on performance of the polarizing M–Z interferometer (Fig. 1). Semiconductor laser (PDL808) produces 50 ps long pulses at 1550 nm with a certain polarization state. The phase modulator is a LiNbO3 -based waveguide that can be driven up to 3 GHz. Such fast modulation implies possibility of high bit rate. PM’s driving voltage (Vo) is tested at 4.8 v. Fibers between PBS1 and PBS2 and between both PBSs and PM are all polarization-maintaining and their polarization extinction ratio is tested above 25 dB. High extinction ratio is necessary in order to ensure good polarization-interference. Both arm lengths of the interferometer (PM’s length included) are about 2 m with a slightly difference less then 1 mm between each arm. Two points are adjustable: One is the PM which introduces the phase shift between arms of the interferometer; the other is the PC which controls the input state. To obtain the four states discussed in Table 1 experimentally, input state need not be strictly 45◦ linearly polarized, but be a state of which the two orthogonal linear components (|H  component that goes into path II and |V  component that goes into path I) are of the same intensity on output end of the interferometer (PBS2) when they interfere. In this case, output state varies to different PM input voltages. Existing phase difference between arms of the interferometer is −Φo when PM introduce no phase shift; 45◦ linearly polarized, 135◦ linearly polarized, left-circular polarized or rightcircular polarized are obtained when PM voltage is Vo∗ Φo /π , Vo +Vo∗ Φo /π , 3Vo /2+Vo∗ Φo /π , and Vo /2+Vo∗ Φo /π , respectively. Optimum extinction ratio of the 45◦ linearly polarized and 135◦ linearly polarized pulse are test at the optimization of 21.96 dB. Our experiment then implements a slightly simplified version of the free-space system on an optical table. Encoded qubit out of the Alice’s interferometer is directed into a single-mode (SM) optical fiber for delivery to transmitting optics, and emitted towards Bob’s receiver. At Bob’s side, the qubit is collected by receiving optics and recovered by PC to the initial polar-

ization state, and then directed into the polarization analysis and detection system. The coupling efficiency between transmitter and receiver for 20 centimeter transmission distance is 46.7%. When PM modulates from −2Vo to 2Vo, the visibility of the interference fringes showed at output 1 and output 2 is 96.1%. This implies a 1.95% bit error rate for QKD sifted bits; however, this would be tolerated in the QKD because of the absolute security of the bits. And we conclude that the dominant contributions to the BER are from optical misalignment and intrinsic imperfections of the polarizing elements. BER can largely be lower by improved optical quality of polarization-maintaining fiber and PBSs. Thanks to the use of M–Z interferometric structures to generate and analyze polarization states, polarization coding and decoding speeds are only limited by the PMs’ modulation speeds, which is up to 3 GHz. The system can keep stable as long as tens of minutes up to half an hour. Phase adjustment can be used to compensate the phase drift in the interferometer hence keeping the system stable. In summary, we have demonstrated phase-modulation polarization encoding system for free-space QKD. The most fascinating point of this system is that limits on coding speed are only determined by performance of the phase modulators, therefore it may be as high as GHz. This would has a potential to support free-space QKD with high bit rate. Based on the structure of polarizing M–Z interferometer, only two detectors and one laser are needed to implement BB84 protocol. Randomly selected polarization states are produced and their complementary bases are achieved at different input voltage of the phase modulator, within a voltage shift within 2Vo (Vo equals to 4.8 v in our system). Linear polarized state is test at 21.96 dB extinction ratio. If achieved by integrated optical components, such as the Mach–Zehnder Germano-silicate waveguide interferometers [12], this phase-modulation polarization coding scheme can be more stable and higher extinction ratio can be get. As a result, this scheme quite suits for the practical realization of encoding qubits in free-space QKD.

X. Liu et al. / Physics Letters A 358 (2006) 386–389

Acknowledgements This study has been funded by National 973 project: G2001039302; Guangdong 2003A103405; Guangzhou 2001Z-095-01. References [1] C.H. Bennett, F. Bessette, G. Brassard, L. Salvail, J.A. Smolin, J. Cryptology 5 (1992) 3. [2] W.T. Buttler, R.J. Hughes, P.G. Kwiat, S.K. Lamoreaux, G.G. Luther, G.L. Morgan, J.E. Nordholt, C.G. Peterson, C.M. Simmons, Phys. Rev. Lett. 81 (1998) 3283. [3] W.T. Buttler, R.J. Hughes, S.K. Lamoreaux, G.L. Morgan, J.E. Nordholt, C.G. Peterson, Phys. Rev. Lett. 84 (2000) 5652. [4] J.G. Rarity, P.M. Gorman, P.R. Tapster, Electron. Lett. 37 (2001) 512.

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